A differential equation for undamped forced non-linear oscillations. II

Extract

In a previous paper of the same title (1) on

where e(t) is even and has least period 2π, I have shown that the equation has a family of periodic solutions or, equivalently, that a topological transformation T has a family of fixed points. Since much of the detail in (1) is irrelevant here, it will be convenient if the results we must quote about are put into the following form:

Theoerm A. There is a constant integer p such that, whenever k ≥ p, there is at least one solution x_k(t) of (1·1) for which

(i) x_k(0) > 0, ẋ_k(0) = 0 and x_k(t) has period 2π;

(ii) x_k(t) = x_k(2π–t);

(iii) x_k(t) has, in 0 ≤ t < 2π, k positive maxima, k negative minima and no other stationary points;

(iv)

This except (ii), is a special case of Theorem 2 with the notation x*(t|k, 1, 0) simplified to x_k(t) and the constant p introduced to guarantee that (vi) of Theorem 2 implies (iv) above. Finally Lemma 1 of (1), in its original form or as specialized in Theorem B (i) below, gives (ii).

We shall write a_k = x_k(0) and note, though we do not need to use it here, that, by (v) of Theorem 2, a_k = kw + O(k⁻²) as k→ ∞. If F_k denotes the point (a_k, 0) of the (a, b)-plane then F_k is a fixed point of T and we can best state this paper's results in terms of the segment joining a pair of such points.